Besides their similarities, there exist a key difference between these two geometric structures. The geometry of the quantum computation addresses the problem to reach a point efficiently by local operations. So it focuses on the computability of a physical system. For more discussion on the computability and physical systems, please refer to [1]}. On the contrary, in the template matching problem, we do not emphasis to use local operations. The reason falls in that the deformation curve \(g(t)\)  is in fact generated purely by local operations since it's an integration curve of a time-varying velocity field, which is essentially local. Also in the template matching problem, it seems that we do not care if a transformation \(g \in Diff(M)\)  can be efficiently computed.
